More

Here is the single compact formula for the example:

\boxed{
f_{n+1}(x)=
\tanh\!\left[
\left(3.7+0.3r_n(x)\right)
\sin\!\left(
2^{n+1}\pi x+f_n(x)
\right)
\right]
}

with the regularity feedback

\boxed{
r_n(x)=
\tanh\!\left(
\alpha\,
\frac{|f_n'(x)|}{1+|f_n''(x)|}
\right).
}

Start with, for example,

\boxed{f_0(x)=\sin(2\pi x),\qquad 0\le x\le1.}

So the complete system is

\boxed{
\begin{cases}
f_0(x)=\sin(2\pi x),\\[2mm]
r_n(x)=
\tanh\!\left(
\alpha\dfrac{|f_n'(x)|}{1+|f_n''(x)|}
\right),\\[3mm]
f_{n+1}(x)=
\tanh\!\left[
(3.7+0.3r_n(x))
\sin(2^{n+1}\pi x+f_n(x))
\right].
\end{cases}}

This gives boundedness + nonlinear feedback + rapidly increasing scales + potentially chaotic dynamics in one compact construction.